Models for the Dynamics of Hyperbranched Macromolecules Alexander Blumen, * 1 Aurel Jurjiu, 1 Th. Koslowski, 2 Ch. Friedrich 3 Summary: We focus on the motion of hyperbranched macromolecules in solution, paying particular attention to the relation between underlying topological structure and dynamics; we consider especially the mechanical moduli. Under the prominent representatives of hyperbranched polymers are both regular structures (such as the dendrimers) as well as disordered structures (such as irregular Cayley-trees). Evi- dently, batch-prepared hyperbranched macromolecules are closer to the latter. In order to theoretically determine their mechanical moduli we employ the method of generalized Gaussian structures (GGS), which allows us to study the situation including or excluding the hydrodynamic interactions (HI). Disordered hyperbranched structures display a complex dynamics; here we recall several analytical and numerical schemes for determining it and compare our theoretical results to the experimental data. Keywords: Cayley-trees; dendrimers; fractals; hyperbranched macromolecules; scaling Introduction Much recent interest has developed around hyperbranched macromolecules [1,2], structures without loops and hence (topo- logically-speaking) tree-like. A well-known subclass of such tree-like molecules are, of course, the dendrimers [3,4], whose con- stitutive pattern is extremely regular. Evidently, the synthesis of perfectly regular dendrimers is by far more demanding than that of usual hyperbranched macromole- cules, for which in batch reactions one accepts a certain polydispersity and also a high pattern diversity. Furthermore, besides the dendrimers, also other regular hyperbranched structures are possible; one such family is given by the regular hyper- branched fractals (RHF) [5,6]. In many applications not only the static but also the dynamical properties of macromolecules matter. Here we will focus on the dynamics of hyperbranched macro- molecules, and particularly on their mechanical properties, given by the mechanical moduli G 0 (v) and G 00 (v). We will perform our analysis in the framework of generalized Gaussian structures (GGS), which are extensions of the Rouse- and Zimm-models to arbitrary topologies, see a recent review [7]. From the beginning we find typical differences in the response functions according to which structure we are con- sidering. While for dendrimers the response functions do not scale [8–10], i.e. do not depend algebraically, as a power law, on time or frequency [11], a different behavior arises for special classes of hyperbranched structures. We recall that RHF do scale [5,6], in a way quite similar to the well- known behavior found for linear macro- molecules. The advantage of focusing on RHF is that their eigenfrequencies can be readily computed to very high accuracy using recursion formulas. This fact dis- penses us in the Rouse-picture from having to diagonalize very large matrices; in Macromol. Symp. 2006, 237, 53–59 DOI: 10.1002/masy.200650506 53 1 Theoretical Polymer Physics, Hermann-Herder-Str. 3, 2 Institute for Physical Chemistry, Albertstr. 23 a, 3 Materials Research Center (FMF) and Institute for Macromolecular Chemistry, Stefan-Meier-Str. 21 and 31, University of Freiburg, D-79104 Freiburg, Germany ß 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim